"This chapter includes nearly all that is known about defining equations for secant varieties of Segre and Veranese varieties …"

At the conclusion of the meta thread that was started by Andy Putman (who wondered why this question was being downvoted) concrete examples now are given of higher-order 'almost-Hilbert' varieties. The following intent is posted there:

Last night I discovered a brand-new monograph by Joseph Landsberg that encompasses more-or-less the answer sought, and so I have amended the beginning of the question to be a pointer to Landsberg's monograph.

Sometime in the next week or two I will post a concrete mathematical question — framed within the context that Landsberg's monograph supplies — asking for a classification of all multilinear varieties having unit-dimension defect with respect to their natural Segre embedding.

At that time I will request closure of the original question, to be supplanted by this classification question.

Please let me apologize for the deficiencies of original question in conveying to MOF readers the mathematical depth and breadth of these multilinear varieties, their implications for fundamental quantum physics, and their numerous practical applications in engineering fields ranging from the computational complexity matrix multiplication to the simulation of quantum transport.

Fortunately, Joseph Landsberg's recent writings have done an immensely better job of this than the original draft of my MOF question did!

My appreciation and thanks are extended to all who have provided comments, and in particular, sincere congratulations are extended to Theo Johnson-Freyd for providing an answer that has received MOF's first-ever Gold Reversal Medal. It has been great fun to help this happen!

The question asked (as clarified per Joseph Landsberg)

For the theorem of algebraic geometry that is specified below, please provide a reference (or references) that:

states the theorem rigorously,

proves the theorem explicitly,

within a framework that extends naturally to multi-linear algebraic varieties

The theorem for which mathematical references are sought

Let $k\ge1$ be an integer and let $\boldsymbol{\psi}=\{\psi_{(mn)}\}$ and $\boldsymbol{\xi}=\{\xi_{(srm)}\}$ be vectors in $\mathbb{C}^{k^2}$ and $\mathbb{C}^{2(k-1)k}$ respectively. Here ${(}\dots{)}$ is a multi-index, repeated indices are summed, and the indices $\{s,r,m,n\}$ range over $s \in \{1,2\}$, $r \in \{1,\dots,k-1\}$, and $m,n \in \{1,\dots,k\}$. Then we have:

Context of the question in multilinear algebraic geometry

Primary consideration should be given to references that prove the theorem and/or discuss related theorems within a mathematical framework that extends naturally to multilinear algebraic varieties.

Secondary consideration should be given to references that are reasonably accessible to the (many) engineers and physicists for whom these multilinear varieties increasingly are finding practical applications.

In the context of algebraic geometry, $r$ may be regarded as an index over $(k{-}1)$ order-2 Segre varieties that enter in a rank-$(k{-}2)$ secant join having the natural Segre embedding in the tensor product space $\mathcal{H}_1 \otimes \mathcal{H}_2$.

In the notation of Joe Harris' Algebraic Geometry: a First Course, the theorem asserts the identity of the preceding Segre embedding with what is called the generic determinantal variety $\mathcal{M}^{(kk)\!}_{k{-}1}$ that comprises (by definition) the set of $k\times k$ complex matrices having matrix rank $k-1$.

Attention is directed particularly to a passage in Harris (page 100) that states:

"We should draw a fundamental and important distinction between bi- and tri- or multilinear objects […] whose invariants are far from being completely understood."

Thus, although the theorem stated can be solved via specialized techniques that apply uniquely to bilinear varieties, a broader and deeper grounding is sought for this theorem within the context that modern algebraic geometry provides, with regard especially to techniques that extend naturally to generic multilinear algebraic varieties. To borrow a phrase from Richard Hamming, "The purpose of the question is insight, not theorems."

Quantum physics and engineering applications

In the context of quantum physics, $s$ may be regarded as an index over two $k$-dimensional Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, each equipped with an $k$-element orthonormal basis, such that $(mn)$ is a multi-index over the quantum amplitudes $\psi_{(mn)}$ that are naturally associated to the $k^2$ orthonormal basis vectors of the bipartite Hilbert space $\mathcal{H}_1 \otimes \mathcal{H}_2$.

In quantum systems engineering (QSE), determinantal varieties are the bread-and-butter state-spaces of large-scale quantum simulations, because they support both the natural geometric pullback of thermodynamical relations and conservation laws and the numerically efficient integration of dynamical trajectories that respect these relations.

The origin of the SHLT name

The name "SHLT" is a homage to the following dialog line in the film Second Hand Lions:

The word defective refers specifically to a $\mathcal{M}^{(kk)\!}_{k{-}1}$ determinantal variety's one-dimensional (nonlinear) rank-defect as a quantum state-space, relative to the $k^2$-dimensional Hilbert space in which it is immersed (see below).

As a starting-point, the SHLT is mentioned — but regrettably only in passing and with no derivation given — in the paragraphs following Example 9.2 on page 99 of Harris.

There exists a finite integer $k\lesssim 2^5$ such that no practicable quantum experiment can observationally disprove the hypothesis that the state-space of a symmetrically bipartite dynamical system is $k$-Lion rather than Hilbert.

Physically, the limit $k\lesssim 2^5$ corresponds to the case of quantum entanglement in a bipartite dynamical system having $5+5=10$ qubits in total.

The Weak $k$-Lion Hypothesis is sufficiently difficult to test — although by construction the required tests are far easier than demonstrating fault-tolerant quantum computing (FTLC) — that it is reasonable to suppose that Weak $k$-Lion Hypothesis cannot be feasibly be disconfirmed even for for quite small values of $k$. Hence it is both mathematically and physically natural to conjecture:

The Strong $k$-Lion Hypothesis

There exists a constant of Nature in the form of a finite integer $k$, such that no experiment can observationally disprove the hypothesis that the state-space of a symmetrically bipartite dynamical system is $k$-Lion rather than Hilbert, for the fundamental reason that the dynamical state-space of Nature is a determinantal variety rather than a Hilbert space.

This question's threefour five-level reward structure

Associated to this question is a threefour five-level reward structure:

@Andres Calcedo: the question is simply difficult to comprehend - it either reduces to a trivial linear algebraic fact (see Theo's answer) or involves some unspecified "algebraic constraints" which are not at all clear from the question. Plus, a lot of "background" (thermodynamics, quantum varieties, talk about pullbacks) makes it even more muddled, the relation to quantum computing is completely opaque (even for me, who does research in quantum computing), and frankly speaking, sentences with "STEM" being every other word don't seem to hep.
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Marcin KotowskiFeb 12 '12 at 0:31

6

In short, the question would be much better if OP cut all the hype about STEM, lions, $100,000 rewards, scalable quantum computing etc. and left the underlying mathematical question clear and well defined (in particular, it should be obvious why the question is not trivial).
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Marcin KotowskiFeb 12 '12 at 0:35

Please let me say that I consider the various comments on my question to have been (all of them) very valuable, and so later today (Sunday) I will post an amended version that attempts to address these comments all-at-once. Especially I appreciate and am grateful for Theo Johnson-Freyd's comments---as pertaining both narrowly to notation and broadly to mathematical context---and it has been exceedingly enjoyable to watch this question help Theo (albeit inadvertently) earn MOF's first Gold Reversal Medal. So thank you all very much!
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John SidlesFeb 12 '12 at 18:42

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What could be very helpful for understanding the question is a simple overview of the background and down to earth description of the conjecture. It looks that John is referring to a method, (perhaps even a method used in practice) to replace a large Hilbert state space of a quantum system by a much lower dimensional algebraic variety of some sort. Taking this variety as a the space of states you can still (but I dont understand how) discuss quantum-like evolutions. John's conjecture is that such much lower dimensional approximation give a good approximation to practical quantum evoltions.
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Gil KalaiFeb 13 '12 at 22:27

Hm. But the displayed statement of Theo is something that anyone who's taken a good course on linear algebra should be able to prove... Surely STEM professionals have taken a linear algebra course, no?
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Mariano Suárez-Alvarez♦Feb 10 '12 at 22:12

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The linear algebra course doesn't even have to be that good - I put problems like that on my matrix-algebra-for-engineers exams and the decent students usually get them right.
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Paul SiegelFeb 11 '12 at 3:49

7

@John: One thing that I think is hard for the mathematicians here to process is that you keep referring to Harris Algebraic Geometry, which is a GTM and not in an easy subject. I might have misunderstood what you mean the second-hand lion theorem to mean, but my understanding of it is a theorem we teach in second-year calculus to every STEM major. (It is somewhat disappointing to hear that they so quickly forget it!) Conversely, typically 100 pages of a GTM is most of a semester of a graduate-level class. For example, I have never read Harris, nor any textbook on algebraic geometry.
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Theo Johnson-FreydFeb 11 '12 at 4:04

6

I'm giving a +1 to Theo for a frivolous reason: to bring him closer to a gold reversal badge. (I haven't thought hard whether it completely answers the question, but the last sentence does seem reasonable to me.)
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Todd Trimble♦Feb 11 '12 at 20:03